Ever since Pătraşcu's [1]} 3SUM-hardness for the all-edge and listing versions of Triangle, it has been a pressing open question to prove the same for detection.
APSP has been connected to Triangle detection in the work of Vassilevska Williams and Williams [2]} but only in a restricted sense: the two problems are subcubic-equivalent for combinatorial algorithms in dense graphs.
Extending such results to general algorithms or to sparse graphs is a well-known challenge.
